Programs for developing FELs using the laser wakefield accelerator (LWFA) technology exist all over the world [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}. So far, there has been a single experimental demonstration of this concept at soft x-ray wavelengths [7]}. The main challenge in adopting the LWFA technology for FELs is yet insufficient quality of the accelerated beams [9]}, [10]}. The accelerated electrons in LWFAs typically originate from the background plasma [11]}, [12]}, [13]}, [14]}, [15]}, [16]}. That naturally does not allow for much control over injection. As a result, injection typically happens over an extended distance rather than instantaneously and the resulting electron bunch has large energy spread.
